Ph.D. Thesis - Physics

Figure 5-15: Plot of the motional coupling rate ωex as a function of the ion-ion spacing d in
lattice and linear ion traps. For the lattice trap, values at d=1 mm are extrapolated from
the lattice trap data in this chapter, while for the linear trap d can readily be calculated
given a frequency ω. We see that for any experimentally feasible set of values, including d,
ωex is significantly weaker in the lattice trap than in the linear trap.
tightly-focused laser beam, and arises from a spatially-dependent AC Stark shift. For a
5 W beam of 532 nm radiation that is focused from 50 µm to 3.5 µm over a distance of d =
50 µm, in traps operating at ω = 2π·250 kHz, we calculate a J coupling of 10 3 s −1 , which
should be observable if the dominant decoherence time is significantly greater than 2π/J.
Similar values can be obtained by using less powerful lasers closer to the atomic resonance;
we use the 532 nm beam as an example only because of the readily-available solid-state
lasers at this wavelength.
The motional decoherence rate expected in microfabricated surface-electrode traps be-
comes small relative to the internal state decoherence time if the trap is cooled to 6 K;
rates for the former have been measured at as low as ˙n = 5 quanta/s [LGA + 08]. Internal
state decoherence times depend on the specific ion being used and also on classical controls,
but coherence times as long as T = 10 s have been reported [LOJ + 05, HSKH + 05]. The
important point is that both internal and motional decoherence channels are much slower
than the J-coupling: T > 1/˙n ≫ 1/J.
Unfortunately, the scaling properties of lattice traps do not favor such a low secular
frequency at small ion-ion spacings. The fact that ω ∝ 1/d is again problematic for the
coupling rate. According to Eq. (5.9), J actually increases linearly with d, a result also
noted in Ref. [CW08]. In Fig. 5-16, we plot a comparison of the J-coupling rates in lattice
and linear traps.
Greatly increasing the trap size is not only impractical, but renders the width of the
ground state wave function of each trapped ion comparable to the laser wavelength, leaving
the system outside the Lamb-Dicke confinement regime. Fig. 5-16 illustrates how the J
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